The Algebra Seminar
Spring 2009
The seminar usually meets Tuesdays in room LN 2205 at 2:50 p.m. There will be refreshments in
the Anderson Reading Room at 4:00.
Organizers: Alex Feingold and Adrian Vasiu
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January 26 : Organizational meeting
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February 3 : Viji Thomas
Title: An introduction to generalized nonabelian tensor product and examples.
Abstract: R. Brown and J. L. Loday first introduced the non-abelian tensor product $G\otimes H$ for groups $G$ and $H$ in context with an application in homotopy theory. Let $G$ and $H$ be groups which act on each other via automorphisms and which act on themselves via conjugation. The actions of $G$ and $H$ are said to be compatible, if $^{^h g}h'=\; ^{hgh^{-1}}h'$ and $^{^g h}g'=\ ^{ghg^{-1}}g'$ for all $g,g'\in G$, $h,h'\in H$. The non-abelian tensor product $G\otimes H$ is defined provided $G$ and $H$ act compatibly. In such a case $G\otimes H$ is the group generated by the symbols $g\otimes h$ with relations $gg'\otimes h=(^gg'\otimes \;^gh)(g\otimes h)\;$ and $g\otimes hh'=(g\otimes h)(^hg\otimes \;^hh')\;$, where $^gg'=gg'g^{-1}$ and $^hh'=hh'h^{-1}$. In this talk I will present a generalization of the above construction.
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February 5 (colloquium of interest to algebraists, Thursday 4:30 -- 5:30 p.m.) : Dmytro Savchuk, Texas A&M University
Title: Graphs related to Thompson's group F
Abstract: Perhaps, the most intriguing open question about Thompson's group F is whether or not it is amenable. We approach this question from two different points of view.
On the one hand we explicitly construct the Schreier graphs of Thompson's group F with respect to the stabilizers of each irrational or dyadic rational point of the interval [0,1] and the standard generating set {x_0, x_1}, and show that these graphs are amenable.
On the other hand we describe the structure of an induced subgraph of the Cayley graph of F with respect to the generating set {x_0,x_1}, containing all vertices of the form x_nw for w in the monoid generated by x_0 and x_1 and n>=0. We show that this graph is non-amenable.
Unfortunately, none of the above approaches gives the answer to the ultimate question about the amenability of F, but both shed some light on the structure of the group itself.
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February 10 : Marcin Mazur
Title: Probability and generators of rings.
Abstract: will discuss a joint work (in progress) with Bogdan Petrenko, where we apply probabilistic methods to prove various results about generators of rings whose additive group is free of finite rank.
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February 17 : Ilir Snopce
Title: Uniform groups, $p$-adic analytic groups and their Lie algebras
Abstract: In this talk I will describe two different ways of associating $\mathbb{Z}_p$-Lie algebras to uniform groups. If time permits, I will also discuss $p$-adic analytic groups and their associated Lie algebras.
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February 24 : Ryan McCulloch
Title: Recognition of Matrix Rings
Abstract: We look at some conditions for a ring R to be a complete n by n matrix ring and apply these. This is a continuation of a talk from last
semester based on work by J.C. Robson.
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March 3 : Martha Kilpack
Title: Join semi-distributive property of the lattice of closure systems on a
finite set.
Abstract: We will take a look at lattices, in particular lattices that have the
property of being "lower bound" (not in the usual definition). We will
consider what makes them join semi-distributive. Finally, we will apply
this to the lattice of all the closure systems of a finite set S.
The talk is based on a paper by Nathalie Caspard and Bernard Monjardet of
Universite Paris and a book "Free Lattices" by R. Freese, J. Jezek, and
J.B. Nation.
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March 10 : Jinghao Li
Title: Nagata rings
Abstract: We will investigate basic examples and properties of Nagata rings.
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March 17 : Donald Passman, University of Wisconsin-Madison
Title: Invariant ideals in a commutative group ring
Abstract:
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March 19 (colloquium of interest to algebraists, Thursday 4:30 -- 5:30 p.m.) : Donald Passman, University of Wisconsin-Madison
Title: What is a group ring?
Abstract:
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March 24 : Xiao Xiao
Title: Unipotent group schemes
Abstract: I will present the concept of a unipotent group scheme over a field k. If k has characteristic p, then we will show that the truncated Witt vector group schemes W_n are unipotent.
If k is algebraically closed, I will also present a couple of results on the classification of commutative smooth unipotent group schemes over k.
The accent will be on the comparison between results over the complex numbers and results in positive characteristic.
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March 31 : Quincy Loney
Title: Constructing affine Lie algebra representations using Clifford algebras
Abstract: We will discuss the construction of representations of the
special orthogonal affine Kac-Moody Lie algebras using infinite dimensional Clifford
algebras. As in the finite dimensional cases discussed in Fall 2007, we find analogs
of the adjoint, natural and spinor representations.
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April 7: No meeting
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April 14, 2009, 1:15 - 2:15, LN-2205
(Cross Listing from the
Combinatorics Seminar) : Luise-Charlotte Kappe
Title: On the covering numbers of symmetric and alternating groups:
An exercise in combinatorial optimization
Abstract: Any finite non-cyclic group is the set-theoretic union of finitely
many proper subgroups. The minimal number of subgroups needed to cover the
group is called the covering number of the group. It is well known that no
group is the union of two proper subgroups and it is less well known that
already Scorza showed in 1926 that a group is the union of three proper
subgroups if and only if it has a homomorphic image isomorphic to the
Klein-4-group. The question arises what is the covering number of a given
group and what values n can occur as covering numbers. Tomkinson showed that
the covering number of any solvable non-cyclic group has the form prime
power plus one. In addition he showed that there exists no group with
covering number 7 and conjectured that there are no groups with covering
numbers 11, 13, and 15.
In this context it is of interest to determine the covering numbers of
non-solvable and in particular simple groups. So far very little is known
and otherwise very accessible groups,like alternating and symmetric
groups, pose a particular challenge. We determine the covering numbers of
some small alternating and symmetric groups. This is an exercise in
combinatorial optimization and some of the results have been obtained with
the help of GAP. This is joint work with Joanne Redden of Elmira College.
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April 14 : Terry Gannon, University of Alberta, Canada
Title: Galois actions on finite groups
Abstract: A lesser known but fundamental symmetry of character tables
of finite groups is the Galois action. I explain how this is the first room in
an infinite tower of Galois actions on finite groups.
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April 16 (colloquium of interest to algebraists, Thursday 4:30 -- 5:30 p.m.) : Terry Gannon, University of Alberta, Canada
Title: From the trefoil knot to modular forms
Abstract: In my talk I will try to identify the actual underlying
symmetry of modular forms. I'll argue that braid groups are more fundamental than the
obvious answer, e.g. the modular group SL(2,Z). I'll give some simple
applications and examples. Little in the way of background will be assumed
(for example I'll also explain what modular forms are all about)
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April 21 : Dandrielle Lewis
Title: Classification of Extraspecial p-groups (Part I of two talks for Candidacy Exam.)
Abstract: In this talk, I will introduce vector spaces endowed with forms and extraspecial
p-groups. Then we will look at a connection between them. Using this connection, we will then able to
give a classification of extraspecial groups.
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April 23 (Time: 1:15 PM, Place: LN-2201) : Dandrielle Lewis
Title: The Automorphism group of an Extraspecial p-group (Part II of Candidacy Exam.)
Abstract: We will use our classification of extraspecial groups and a couple of other
results dealing with isometries to determine the automorphism group of an extraspecial group.
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April 28 : Ding Ding
Title: Descent theory for modules
Abstract: In this talk we study the following question: given a ring homomorphism R\to S, when an S-module M is of the form N \otimes_R S for some
R-module N? We show that if S is faithfully flat over R, then R-modules are
equivalent with S-modules equipped with a descent datum (i.e., with a suitable map that allows to recapture N from M).
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May 5 : Adam James Perry
Title: Groups with directly decomposable subgroup lattices
Abstract: It will be shown that groups with directly decomposable
subgroup lattices are exactly the groups that are direct products of groups of
co-prime order. It will follow given a finite lattice L with no chain
as a direct component, there are only a finite number of finite groups
whose lattice is L.
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May 19 : Xiao Xiao
Title: Basic properties of F-crystals. (Part I of two talks for Candidacy Exam.)
Abstract: An F-crystal over a perfect field k is a pair (M,phi),
where M is a free module of finite rank over the ring of Witt vectors of k and where phi is an injective Frobenius-linear endomorphism of M.
We will discuss Hodge polygons and Newton polygons of F-crystals and present some fundamental results pertaining to them.
For instance, when k is algebraically closed, we present the Dieudonn\'e and Manin classification
of F-crystals up to isogeny (i.e., of those pairs obtained from (M,phi)'s by inverting p). In the last part, we will introduce some invariants of F-crystals (some new and some old), which are important in their study up to isomorphism, and provide some upper bounds of them.
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May 20 (Wednesday, regular time 2:50--3:50) : Xiao Xiao
Title: Basic properties of F-crystals. (Part II of two talks for Candidacy Exam.)
Abstract: See above.
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June 4 and 5 (Thursday and Friday, special time 1:00--2:30) : Quincy Loney
Title: Representations of affine Kac-Moody Lie Algebras (Parts I and II of two
talks for a Candidacy Exam.)
Abstract: We will discuss the (bosonic) construction of representations of
the special orthogonal affine Kac-Moody Lie algebras g^ = so^(2n) using
Heisenberg algebras h^. We will also discuss the construction of those g^
representations using (fermionic) Clifford algebra representations from
previous lectures. In particular, we compare the adjoint, natural and
spinor representations for both constructions.
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